Before analyzing data statistically, it is important to consider if
the data were collected appropriately. Many years of labor and even
careers have been virtually wasted because of fundamental flaws in the
data collection step. The statistical analysis will only likely be a
minor part of the total expense of a properly conducted experiment,
so time, effort, and money spent ensuring the data are collected
appropriately is certainly well spent. The computer adage
Garbage In, Garbage Out or GIGO is
rather apropos.

Ensure that the sample size is large enough.

Although a large sample is no guarantee of avoiding bias,
too small a sample is a recipe for disaster.
How to determine minimum sample size will be at least touched on in
lesson 11 (Hinkle chapter 13).
There are well established techniques to determine such.
These techniques are based on the Central Limit Theorem discussed
later in this lesson.

Better results are obtained by measuring instead of asking.

A good classroom example would be to collect people's heights. We
expect heights might be normally distributed.
Asking will result in several sources of error.
Perhaps the most common being exaggeration, rounding, hair style,
and shoe heel variation or even complete absence of shoes.
The units of measure: inches; feet and inches; or centimeters
isn't obvious either.
Were you instead to measure each individual, these sources of error
could be reduced. You may still encounter systematic errors.
Following are some sources of systematic error.
Perhaps your measuring device is defective. Specific examples might
include the common fact that rulers often don't start exactly at zero,
but have a little extra margin. Maybe the measuring tape is marked off in
inches on one side and tenth's of a foot on the other and sometimes the
wrong side is read.
Tape measures can become kinked or even tangled
(especially surveying caves).
Perhaps being a statistics students correlates with being shorter or taller
for some unknown reason. This might only be a problem if you were to
use your data to represent a larger population.

The medium used (mail, phone, personal interview) is important.

Surveys are a very popular method of data collection for social issues.
Mail surveys tend to have a lower response rates which will distort and
hence flaw a sample. Although telephone surveys may be relatively
efficient and inexpensive, the more time consuming and correspondingly
expensive personal interview allows more detailed and complex data to
be collected.
Be not called by telemarketers.

Be sure the sample is representative of the population.

An observational study observes individuals and measures variables
of interest but does not attempt to influence the responses.
An experiment deliberately imposes some treatment on individuals
in order to observe their responses.
Observational studies are then a poor way to gauge the effect of an
intervention. When our goal is to understand cause and effect,
experiments are the only source of fully convincing data.
However, imposing treatments may produce some ethical concerns.
See more below under experimental design.

Before we move on to the next point,
we should note that some studies are retrospective,
or involve looking back at past events,
whereas others are prospective or track groups forward in time.

Sampling is the fundamental method of inferring information
about an entire population without going to the trouble or expense of
measuring every member of the population. Developing the proper sampling
technique can greatly affect the accuracy of your results.

Statisticians have classified sampling into five common types, as follows.

Random
Sampling: Members of the population are chosen in such a way that
all have an equal chance to be measured.

Other names for random sampling include representative
and proportionate sampling because all groups
should be proportionately represented.
Consider what might happen if a telephone directory were used
as a source for randomly selecting survey participants.
Some people have no phone, others have multiple phones and corresponding listings.
Still others have unlisted phone numbers.
In affluent areas unlisted phone numbers may approach half the population!
Now-a-days many are giving up lands lines and use cell phone
exclusively. Cell phone directories are controversial at best.
Pollsters commonly use computers to generate and dial
phone numbers in an attempt to circumvent these problems.
However, many people consider such use of the telephone as
an invasion of their privacy and refusals or hang-ups may well
significantly influence the outcome.
Some of us have learned to recognize these computer dialers and quickly hang up.
Such are the pitfalls which must be carefully considered
in designing an experiment, study, or survey.

Simple random sampling is the least complex and probably the
most widely used sampling technique in behavior science research.
The word simple here differentiates this sampling technique
from other more complex sampling techniques.

A simple random sample or SRS consists of n
elements from the population chosen in such a way that
every set of n individuals has an equal change
of being the sample actually selected.

The statement above is technically true only
if the sampling is done without replacement,
the most common practice.
If done with replacement, each member of the
population has the same probability of being selected.
The difference is slight and subtle and requires
only a minor adjustment.
(The same element could be repeated.)
In either case each sample member is selected
independently of any other sample member.
In dichotomous situations, a fair coin could be flipped.
Names in a hat are another example if there is no selection bias.
For large samples such methods prove cumbersome.
Historically, tables of random digits were commonly used.
Psuedo-random numbers from a computer or calculator
are now more commonly used.

Random sampling must be a structured event to ensure no bias.
These are not haphazardly done, done on the spur of the moment,
or done as a matter of convenience.
In general no randomization no generalization.
Since the point of taking the sample is generally
to generalize the results to the parent population,
the randomization step is extremely important.
A sampling technique which starts out random
may lose such a status as it is processed.
For example, suppose surveys are mailed out
and some recipients fail to return a completed survey.
Worse yet, suppose a mail sack was lost or stolen
so no one from Wyoming even got theirs.

The historic event leading to the word decimate,
where every 10th Roman soldier was killed,
is a gruesome example of systematic sampling.

The reciprocal of the sampling fraction (the ratio of the size of the
sample to the population: (n/N) determines the k (N/n) used.
Once k has been determined, the index of a starting element is
selected within the first k elements by random selection.
The indices for subsequent elements are formed by
adding multiples of k to this starting index.
The sampling continues until we reach the end of the list
at which time n elements will have been selected.
There are technicalities for handling non-integer ks
which add some complexity.

Note how there are now only k different samples possible.
This limitation is often of little consequence when one considers
the difficult procedures often necessary to obtain a simple
random sample and in most situations these samples can be treated
as SRSs as long as no periodic factors bias the sample within
the list. Alphabetic lists of peoples names are generally
free of periodic factors even though family names and
ethnicity (van, Mc) may cause some clustering within the list.

Stratified
Sampling: The population is divided into two or more strata and each
subpopulation is sampled (usually randomly).

Stratum is the singular form of the word strata which means
to spread out. One of the word's most common usage is in geology
to describe the layers of sedimentary rocks which have formed during
the earth's history. Gender and age groups would be commonly used strata.
Classes is another term for strata.
Each stratum must share the same characteristic.
Random sampling may well be used to select a certain number of
data points from each stratum.
This sometimes is the most efficient sampling method.

Stratified sampling takes advantage of some inhomogeneity
(heterogeneity) of a population, whereas random and
systematic sampling generally assume the population is homogeneous.
One often knows how the population is distributed
among these different strata and then usually
proportionally allocates the sample accordingly.
Although this might seem an advantage, the random sampling
process tends to generate similar results.
However, stratified sampling will ensure that no strata are missed.
Stratified random sampling will enhance statisitical
precision, which is a desirable outcome.

Cluster Sampling:
A population is divided into clusters and a few of these
(often randomly selected) clusters are exhaustively sampled.

Exhaustively means considering all elements.
Cluster sampling is used extensively by governmental and private
research organizations. These clusters are naturally formed
groups such as families, classrooms, or even schools.
Hopefully, population elements belong to one and only one cluster.
Multistage sampling is common with cluster sampling.
An example might be a two-stage sample in which
precincts are randomly selected, followed by the random selection
of blocks of residents within these precincts.
All residents then within these selected blocks would be sampled.

Convenience Sampling:
Sampling is done as convenient, often allowing the element to choose whether
or not it is sampled.

Convenience sampling is the easiest and potentially most dangerous.
Often good results can be obtained, but perhaps just as often
the data set may be seriously biased.
Consider collecting GPA information from students in detention.
It may be convenient, but perhaps not representative of the entire student body!

We have listed above several sources of sampling error.
One of the most famous sampling errors occurred in 1948 when
the Gallup poll
predicted Dewey would be elected president over Truman.
The day after the election, such an announcement made the
front page
of a major newspaper!
Gallup then abandoned the quota system and
instituted random sampling based on clusters of interviews nationwide.
Sample subjects should be selected by the pollster.
They should not select themselves as they do via mail or perhaps telephone
surveys. The systematic errors listed above are examples of
nonsampling errors.

Of great debate recently was what to do with the errors which arise in
the decennial US Census.
Considerable time was spent by all three branches of our government
addressing this issue.

Some questions are classified as open,
whereas other questions are classified as closed.
Open questions elicit open-ended responses and thus work best in a
personal interview. Multiple-choice or true/false questions are
a type of closed question. Closed questions can thus more easily
be coded and analyzed by a computer.
If surveys are used be sure to include the survey sponsor,
the date the survey was conducted, the size of the sample,
the nature of the population sampled, the type of survey used,
and the exact wording of the survey questions. Other important
issues include: assessing the risk to those surveyed, the
scientific merit of the survey, and the guarantee of the subject's
consent to participate. An example of risk might be the
hazard of planting ideas (rape, murder, suicide, etc.) in
someone's head or reviving suppressed memories (abuse)
while asking related questions.

More information on experimental design (treatments, factors, blocking,
double blind, latin square,
randomized complete block, matched pairs, replication, and simulation)
should be included here but isn't. Consult any good Statistics book
or take subsequent courses for more information.

A famous example of an experiment is when Benjamin Franklin, famous American
statesman and scientist, determined whether electricity is conducted.
The experiment involved flying a kite in a thunder (and lightning) storm
with a wire from the kite to a key in a bottle. (Don't try this at home!)
(Also,
questions
have arisen as to whether or not he actually performed
this experiment. It seems others did it earlier, only his son may have
been present, and his journals don't support well this event occurring.)
The experimental method is now the basis of the
scientific method.
In statistics we often refer to a random experiment, one for which there
is no way of telling beforehand what the outcome will be.

The act of rolling a fair die, flipping an honest coin, or randomly selecting a card from a deck
are all considered random experiments.

An interesting part of mathematics is the use of common language to describe
mathematical concepts. One such example is the word event. Normally, event
conjures up images of special moments: the prom, banquets, fairs, weddings,
births, .... In dealing with probability, event has a very precise
meaning.

An event is the set of outcomes from a random experiment.
A simple event is an outcome which cannot be broken down.
The sample space is the set of all possible outcomes for a given experiment.

\

T

H

T

TT

HT

H

TH

HH

As indicated above, flipping an honest coin is a random experimentone
has no way beforehand of predicting the outcome.
The sample space is a set which contains all possible outcomes.
For one flip the possible outcomes are heads (H) or tails (T).
For one flip the sample space contains only these two outcomes.
For two flips the four possible outcomes are HH, HT, TH, or TT.
Thus the sample space is {HH, HT, TH, TT}, containing four elements.
Notice the difference between the events HT (heads first) and TH (tails first).
The outcome of a single flip is a simple event, whereas
the outcome from more than one flip is a compound event.

Rolling a standard six-sided (fair) die once would have
a sample space with six outcomes: 1, 2, 3, 4, 5, and 6.
Rolling a pair of dice would have a sample space of
six times six (62) or 36 possible outcomes.
In the activity for lesson 2
we constructed the sample space of rolling a pair of dice
and plotted the distribution of the sum of pips
(See Hinkle, Figure 7.3).

For some interactive web sites involving
rollingdices,
flipping
or spinning coins check out these links.
Be forewarned, however, that if
cards
or a roulette wheel are involved your internet search
is likely to lead you to gambling sites (casinos) whose legality on the
web has been and is being challenged due to its
addictive nature and
those many lives which have been ruined thereby.

The term probability is often used fairly casually and as such
can be rather subjective. The probabilities which form the basis
of inferential statistics are based instead on mathematical
concepts and theory.

Probability is denoted by P and specific
events by A, B, or C.
The shorthand notation used to indicate the probability that event B
occurs is P(B).

Empirical (Experimental) Definition of Probability:P(A) =
number of times A occurred divided by the times the experiment was repeated.

Classical Definition of Probability:P(A) =
number of event A outcomes divided by the size of the sample space.

The probability of something occurring is related to its frequency.
Specifically, when a coin is flipped twice in succession,
in 1 of the 4 possible outcomes heads appeared both times.
Thus the probability was ¼ or 0.25.
It is important to remember that the probability
of A occurring is less than or equal to one.
We have tacitly assumed here that the probability of
heads is equal to that of tails.
Experiments have been conducted to test this.
In such a case, the probability would then be an experimental
rather than a theoretical result.

An event with a probability of 0 is impossible.
An event with a probability of 1 is certain.
0 P(A)
1 for any event A.

Probabilities for random events might be computed exactly.
In such case we express them as fractions.
Other probabilities are obtained by experiment and are thus
approximations which are typically expressed to three
significant digits unless there are compelling reasons for
more or less precision.
Probabilities are often given as percentages.
When doing so be sure to include the percentage symbol (%) since
technically probabilities are always between 0 and 1 inclusive.
For example, certainty corresponds with 100%
and impossibility with 0%.

Probability can be approximated by frequency:P(A) = number of times A occurred divided by
number of times experiment is repeated.

We used the term fair above to describe coins or dies yielding
an equal likelihood for any outcome. Thus a fair coin has a
50% of turning up heads and a 50% chance of turning up tails.
This is often expressed in terms of odds
as 50-50. Each of the two outcomes
is equally likely and thus had a probability of ½.
On rare occasions a coin might
end up on its side, but generally we exclude such events from
the set of outcomes we are considering,
just as we generally consider only the genders of male and female.
We would thus expect a six sided die to have a
1/6 probability
for any face to be on top. Again, the rare chance of balancing
on an edge or corner will generally be excluded, as will be
outcomes where the result cannot be determined (such as the die
falling into a black hole or sewer grate).

If an experiment is repeated over and over,
then the empirical probability approaches the actual probability.

The above statement is often stated as a theorem known as
the Law of Large Numbers.
Determining sample size is an exercise in optimizing tradeoffs in
cost and accuracy.
Large samples should be more accurate but will be more costly,
whereas smaller samples cost less but provide less accuracy.
Those who have not studied statistics tend to scoff at the idea
that a survey of only 1000 (0.001%) people in this country of
100 million voters can give a good estimate of how many
favor a particular candidate or position.
Of course, if your sample is not random, biases will creep in,
and accuracy will suffer.
Later lessons will explore these concepts in greater detail.

Earlier we put two or more simple events together to
create compound events. There are various ways of combining such events.
Specifically, we might ask the number of outcomes when
event AOR event B occurs,
or we might ask the number of outcomes when
event AAND then event B occurs.
The quantity of outcomes will be used as the numerator
when we calculate the probability.

Example: Assume you have 20 M&M® brand candies as follows:
5 orange, 6 yellow, 5 red, and 4 green.
In one selection, how many ways can you select either 1 orange or
1 yellow M&M®? What is the corresponding probability?Answer: Of the 20 M&M's®, 5 are orange and 6 are yellow.
Hence 5+6=11 of the M&M's® are yellow or orange.
The probability of selecting a yellow or orange M&M® is 11/20=0.55.

The M&M's® are either one color or another, hence getting
a certain color is mutually exclusive of getting a different
colorthat is, no M&M's® are rainbow-colored, zebra-striped,
or some shade such as orange-yellow or blue-green which thus might be
judged different colors by different people.
To clarify further the meaning of mutually exclusive, let's say
that only one or another event can occur, never both at the same time.

Example: Assume you have 20 M&M's® color distributed as above.
If selected without replacement, in how many ways can you select
two red ones in two selections?
What is the corresponding probability?Answer: For the first selection, five of the 20 M&M's® are red.
Since we need to get two reds in only two selections, we need only
consider this successful case further, ignoring what happens if
we do not get a red on this first selection.
For the second selection, only four red of the 19 M&M's® remain.
Hence there are 54=20 ways of selecting two reds M&M's®
in two selections. The corresponding probability would be:
(5/20)(4/19)=20/380=1/19 or approximately 0.0526.

The first example above (OR) will be dealt with further below.
We will now discuss the second example (AND then).
We noted above
repeated coin flips and die rolls.
The size of our sample space, that is the set of all possible outcomes,
was the product of the set of possible outcomes for each event:
22=4 for two coin flips and 66=36 for rolling two dice.

If event A can occur in m possible ways
and event B can occur in n possible ways,
there are mn possible ways for both events to occur.

n(A and then B)=n(A)×n(B)

This is generally expressed as event A and then event B occurring.
This is an AND situation where both are performed.
This calculation extends to three or more events.
For example, if event C can occur in o possible ways,
there are mno possible ways for these three
events to turn out.

Example: How many different ways can parents have three children.Answer: For each child we will assume there are only two possible
outcomes (thus neglecting effects of extra X or Y chromosomes, or any other
chromosomal/birth defects). The
number of ways can be calculated: 222 = 8.
These can be listed: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG where
B=boy, G=girl. We could have just as well used the symbols 0 and 1:
000, 001, 010, 011, 100, 101, 110, 111. Note that this is the same
as counting in base 2. This fact can be used to more easily list
outcomes or to check for missing outcomes (exactly 4 have boy first,
exactly 4 have boy second, exactly 4 have boy last, etc).
Another way to represent this information is in tree form with
the branches from each node representing the possibilities for
the next event (see below).
Note that this can become very large and thus listing
or displaying the complete sample space is often impractical.

If event A can occur in m possible ways
and event B can occur in n possible ways,
there are m+n possible ways for either event A or event B to occur,
but only if there are no events in common between them.

n(A or B)=n(A)+n(B)-n(AB)

Because often one works with non-overlapping events, you will find that the
last term is commonly omitted, but added later. It is better to learn the
formula correctly the first time and make a special case when the
intersection is indeed empty.
An empty intersection might occur due to happenstance or it
might occur because the events cannot occur simultaneously,
i.e. the events are mutually exclusive.
In the M&M® example above, the color selections were mutually exclusive.

In the homework you will look at
an example of overlapping events when you calculate
the probability of the green die having a 2 or the red die of having a 5.
A careful inspecation of the diagram in the prior lesson indicates that
although there are six outcomes where the green die has a 2 and six outcomes
where the red die has a 5, we must be careful not to double count
the event where both the green die has a 2 and the red die has a 5.
There are thus only 11 not 12 corresponding outcomes and
the probability was 11/36 or about 0.306.

The factorial rule is used when you want to find the number of
arrangements for ALL objects.

Example: Suppose you have four candles you wish to arrange from left to right
on your dinner table. The four candles are vanilla, mulberry,
orange, and raspberry fragrances (shorthand: V, M, O, R).
How many options do you have?Solution: If you select V first then you still have three options remaining.
If you then pick O, you have two candles to choose from.
You can compute the number of ways to decorate your table by the factoral rule:
for the first choice (event) you have 4 choices;
for the second, 3;
for the third, 2;
and for the last, only 1.
The total ways then to select the four candles are: 4!=4321 = 24.

These types of problems occur frequently and can be summarized as follows.

Another word for arrangements is permutations.
Please recall that the symbol ! is mathematical shorthand for factorial.
n!=n(n-1)! and 1!=1.
Please also note that by definition and because it
makes these types of problems easier, 0!=1.
5! = 54321 = 120,
4! = 24, 3! = 321 = 6, and 2!=2.

Try solving this exercise on your own:
You need to study, practice football, fix dinner, phone a friend,
and go buy a notebook. How many different ways can you arrange your schedule?

Permutation is another name for possible arrangements
with SOME items from a given set. It is
important to remember that order chosen or position arranged is
taken into account.
Hence permutations are similar to
anagrams.
Given below is the necessary equation.

nPr = n! / (n - r)!
where r is the number of items arranged from n elements.

More information on permutations, permutations with repeated
elements, and permutations on a circle can be found at
this location.

Combinations are arrangements of elements without regard to
their order or position.

nCr = n! / (r!(n - r)!)
where r is the number of items taken from n elements.

Note that these numbers are the same as those in Pascal's Triangle,
the binomial formula, and the binomial distribution. Those less than
about four digits become very familiar.

Example: You have five places left for stamps in your stamp book and
you have eight stamps. How many different ways can you select five?Answer: 8!/(5!3!) = 876/(32)=56.
Think of putting them in slots, the first has eight choices,
the next slot has seven choices and so forth as demonstrated.

8 7 6 5 4

Each combination of choosing 5 out of the 8 has permutations of its own.
The five can be arranged in the following ways:

5 4 3 2 1

Thus there are (8!÷3!)÷5! = 8!÷(5!3!)=56 ways to select
five of eight, but 6720 ways to arrange five of eight.

When working with the multiplication rule, keep in mind whether or not
the events are independent.
Independent events are those that do not affect each other.
Otherwise the events are dependent.
When sampling is done with replacement, the selected object
is put back before the next object is selected. The
events remain independent. When down without replacement
the events become dependent.
P(A|B) represents the probability of A
occurring after B has already taken place.
This is known as the conditional probability.

P(A and B) = P(A)·P(B)
if A and B are independent.P(A and B) = P(A)·P(B|A)
if A and B are dependent.

Sometimes the probability of A and B occurring is given,
but the question asks for the probability of B occurring after A.
All that requires is solving the algebraic equation,
P(A and B) = P(A)·P(B|A)
for P(B|A), the conditional probability.

Tree diagrams are a method of double checking your work
when the sample space is small.

Example: A couple plans on having 3 children.
What is the probability of them having two boys and one girl?Answer:

B ---- B ---B

---G

---- G ---B

---G

G ----- B ---B

---G

---- G ---B

---G

2 × 4 = 8

In the chart, there are three different ways to have two boys and one girl.
Thus the probability is 3/8 or 0.375.
One can also think of the only girl being born first, second, or third.
We can do it in a different way: P(GBB) + P(BGB) + P(BBG) =
½×½×½ + ½×½×½
+ ½×½×½ = 1/8 + 1/8 + 1/8 = 3/8.
Of course, those of us who have done this awhile immediately think in
terms of Pascal's Triangle and nCr!

Example: What is the probability of rolling a die twice and
getting two sixes?Answer:P(6)·P(6) = 1/6 × 1/6 = 1/36 = 0.0278.

In Geometry, complementary angles summed to 90°these angles
together complete a right angle.
Another widely used meaning is that complement is opposite, or the
negation of something.
In probability, the complement of event A are
the outcomes which do NOT have event A occurring.
The notation of the complement of A
is a horizontal bar over A (or for these webpages: Ã).
Please note that this spelling and meaning for complement
is distinct from compliment which means a formal expression
of esteem, respect, affection, or admiration.

Example:
A local theater group is planning to give away a season ticket via a raffle.
Eighty women dropped their ticket stubs in the bucket while only 35 men did.
What is the probability of the winning ticket not going to a woman?Solution: Thirty-five men dropped their stubs of the 115 total tickets.
P(not getting a woman) = P(man) = 35/115 = 7/23 = 0.304.

Using the complementary rule with the multiplication rule, one can find the
probability of at least one event being what we want. At least one means
the same as one or more. The complement of one or more is none.
So the multiplication rule is used to find P(none) and then take the
complement of it. P(at least one) = 1 - P(none)!!!

Example:
A person deals you a new five card hand. What is the probability
of having at least one heart?Solution:P(at least one heart) = 1 - P(none) =
1 - 13C039C5 ÷ 52C5 =
1 - (39/52)(38/51)(37/50)(36/49)(35/48)
= 1 - 0.222 = 0.778.
Just think how long it would have taken if instead you calculated
the probabilities for getting
one heart, two hearts...!

Please note, the method used above for computing none is very
general and not well nor widely documented. I'm referring specifically
to the expression:
13C039C5 ÷ 52C5.
This expression is saying of the 13 hearts we choose 0,
whereas of the other 39 cards we choose 5. These two items are multipied
together then divided by the number of ways to choose 5 cards from 52.
Thus to calculate the probability for getting one heart would be:
13C139C4 ÷ 52C5.

As we have seen before, the probability of something
certain to occur (occurring 100% of the time) is one.
Using the addition rule for P(A) and P(Ã),
which are mutually
exclusive because A and Ã cannot occur at the same time and knowing all that is
not in A is in Ã, we end up with P(A)
+ P(Ã) = 1.

P(A) + P(Ã) = 1P(Ã) = 1 - P(A)P(A) = 1 - P(Ã)

Example: A farmer expects to bring 80% of a field of wheat to market.
How much of the wheat is lost by various means of destruction?Solution: 20% is destroyed by mice, drought or other means.
Remember that percentages are equivalent to probabilities: 80% = 0.80 = P(A).
Thus P(Ã) = 1 - 0.8 = 0.2 = 20%.

Thomas Bayes was a 18th century English Presbyterian minister
(and statistician) who said that
probabilities should be revised when we learn more about an event.
Bayesian statistics is very much in vogue and is considered by
some a different "flavor" of statistics.
Bayes' Theorem, also known as Bayes' Rule
gives the solution to what Rev. Bayes called the "converse problem".
Many medical tests give what are known as false positives.
Thus Bayes Theorem is commonly used in paternity suits to calculate
the probability that a defendant really is the father of a child,
given test results which support such a conclusion.
Another example can be found here.

A casino operator generally isn't worried about distribution
of a gambler winnings because he knows that over the long run
the odds favor the casino. Occasionally, someone able to
count cards well might distort the otherwise random nature
of the outcomes. This random nature is dependent on the
cards being well schuffled and multiple decks together makes
counting cards impractical for most humans. Such randomness
underlies the probability upon which inferential statistics
depends.

Seven, eleven, or doubles may get you out of jail
in Monopoly, but a close examination of the 36 possible outcomes
when two dies are rolled and the pips summed indicates these
have probabilities of 6/36, 2/36, and
6/36 for a total of 7/18=0.389.
The corresponding probability for getting out on the second turn
is (7/18)2=0.151.
The corresponding randomness gives variety to the game just as
randomness gives variety to statistical results. However,
there is an underlying distribution which can be analyzed.

The underlying distribution of possible outcomes is important when
we consider the probability of any specific sample. We will briefly
look at a few theoretical distributions which are commonly
encountered.

Consider the combinations examined above and apply it specifically
to the case of selecting six of ten when the ten are
five males and five females. Any number, say x,
between 1 and 5, inclusive, could occur with 6 - x
of the other gender occurring.
However, if we examine the distribution of the
10C6 = 210 different possibilities we discover
5C55C1 = 5
ways five women and one man might be selected,
5C45C2 = 50
ways four women and two men might be selected,
5C35C3 = 100
ways three women and three men might be selected,
and the remaining results are symmetric hence given above.
This is a very leptokurtic distribution (Hinkle Figure 7.4).

Another common underlying distribution is the
binomial distribution which we already
examined in homework 4 problem 1
with the flipping of four coins and gave some
formulae in lesson 4.
This is a special case in the family of binomial distributions
for a given number of trials, where p=q=½.
It is natural to ask what happens when p#q#½.
The same formula as before applies, namely:

Thus, the probability that 5 of the 25 students will be left-handed is about 6%.
As usual, it is important to set up your solution logically.
Carefully identify the important values (n, x, p, etc.)
before cranking out the numbers and presenting your answer.
The TI-83/84 series calculators have BINOMPDF which,
if given the two arguments of n and p, in that order,
will output a list of n+1 probabilities for each value of x,
with the first one being for x=0. BINOMCDF is similar
but gives cumulative frequency. Both are under the 2ndVARS
or DISTR button (entries 0 and A, so you may need to scroll down).
It can be shown that the mean, variance, and standard deviation of a
binomial distribution can be expressed in simple formulae as follows:

mean: =n  p

variance: 2 = n  p  q

std. dev.: =
(n  p  q)

Example: Suppose 20 biased coins are flipped and each
coin has a probability of 75% of coming up heads. Find
the mean and standard deviation for this binomial experiment.Solution:n=20, p=0.75, so q=¼.
=n · p = 20 · 0.75 = 15.
This is as expected, we expect heads to come up about three quarters the time.
=
(n · p · q) =
(20 · 0.75 · ¼) =
3.75
1.936.

Since the binomial distribution tends to become more like the
normal distribution as sample size increases, especially when
p and q are nearly equal, we can often
approximate the binomial using the normal distribution.
More information can be found here
which we will summarize by saying np and nq
must be greater than 10 (or 5 or 15) before this can be done.

The normal distribution is the most important underlying distribution
due to is prevalence in such measurements as intelligence, aptitude,
and achievement. In addition, many of the statistics generated
through inferential statistics are normally distributed or close to
normally distributed.

This lesson's discussion has established a foundation for
the probability and reasonings behind the procedures known
as inferential statistics. Specifically, once we have
taken a sample and measured a corresponding statistic,
we either estimate population parameters or
test hypotheses about these unknown parameters.
One of the most common parameters we wish to estimate
is the population center and the mean is a good measure
of such central tendancy. As it turns out, if the sample
mean is from a random sample, it is a good estimator of
the population mean. To establish how good we need to
examine how the means from all possible samples are distributed.

The center, width, and variability of the
sampling distribution of the mean is determined by
the central limit theorem.

Central Limit Theorem:
As sample size increases, the sampling distribution of sample means
approaches that of a normal distribution with a mean the same as the
population and a standard deviation equal to the standard deviation
of the population divided by the square root of n (the sample size).

Stated another way, if you draw simple random samples (SRS) of size n
from any population whatsoever with mean
and finite standard deviation ,
when n is large, the sampling distribution of the sample means
is close to a
normal distribution with mean
and standard deviation /
(n).
This standard deviation is often called the
standard error of the mean.

It is important to recognize that this standard error of the mean
decreases as sample size increases. This means increased precision
with larger sample size. However, to improve the precision by
a factor of 2 would require an increase in the sample size by
a factor of 4.

A second result is that the shape of the sampling distribution
of the mean resembles more closely a normal distribution as
the sample size increases, even when the population is not normal.

The sampling distribution in the case above of sample means
becomes the underlying distribution of the statistic.
It is an important component in the chain of reasoning
which underpins inferential statistics.
Different sampling distributions will apply to different
sample parameters. The study of inferential statistics
is largely an examination of which distribution applies
to which parameter and developing a familiarity with this
distribution and how to apply an appropriate statistical test.